Polynomials with no zeros on a face of the bidisk

0.44 MB·English

by  Jeffrey S. Geronimo

POLYNOMIALS WITH NO ZEROS ON A FACE OF
THE BIDISK
3
1 JEFFREY S. GERONIMO, PLAMEN ILIEV, AND GREG KNESE
0
2

n
a
Contents
J

5
1. Introduction 1
1
2. Notation and statement of results 5

] 3. Basic orthogonalities of Bernstein-Szego˝ measures 11
V
4. Bernstein-Szego˝ condition implies split-shift condition 15
C
5. Split-shift condition implies Bernstein-Szego˝ condition 21
.
h
6. The matrix condition 25
t
a
7. Description of shift-splits and split-polys 27
m
8. Construction of p from Fourier coefficients 30
[
9. Applications 32

1
9.1. Autoregressive filters 32
v
0 9.2. Full measure characterization 34
1 9.3. Concrete expression for the full measure characterization 36
5
10. Generalized distinguished varieties 38
3
. 10.1. Construction of the sums of squares formula 38
1
0 10.2. Generalized distinguished varieties and determinantal
3 representations 47
1
References 51
:
v
i
X
r
a 1. Introduction
This article is concerned with harmonic analysis and moment prob-
lems as motivated by prediction theory and connections to analytic
function theory and operator theory, continuing a tradition of clas-
sic works such as Helson-Lowdenslager [12], Helson-Szego˝ [13], and
Wiener-Masani [24]. Most of these works are concerned with harmonic
analysis on the unit circle and function theory on the unit disk. In this
Date: January 14, 2013.
JG partially supported by Simons Foundation Grant #210169.
PI partially supported by NSF grant DMS-0901092.
GK partially supported by NSF grant DMS-1048775.
1
2 J. S.GERONIMO,P. ILIEV,AND G. KNESE
article, we work in the setting of the two-torus or bi-circle. Helson-
Lowdenslager [12] was perhaps the first paper to pin down which as-
pectsofharmonicanalysis onthecircleextend inastraightforwardway
to the bi-circle. Factorization of positive trigonometric polynomials is
one area that most certainly does not extend in a straightforward way
from one variable to two, and this topic serves as a good starting point
to motivate the rest of the paper.
The classical Fej´er-Riesz lemma states that a non-negative trigono-
metric polynomial t(θ) in one variable can be factored as
t(θ) = p(eiθ) 2
| |
where p C[z] is a polynomial with no zeros in the unit disk D = z :
∈ {
z < 1 . While this is one of the simplest factorization results it is use-
| | }
ful in signal processing, trigonometric moment problems, and wavelets.
It is also a prototype for more advanced and important factorization
results, such as Szeg˝o’s theorem. A simple degrees of freedom argu-
ment shows that this result cannot be extended without conditions to
two variables. In recent years, progress has been made in extending
this result to two variables. First in Geronimo-Woerdeman [10] a char-
acterization was given of positive bivariate trigonometric polynomials
t that can be factored as
(1.1) t(θ,φ) = p(eiθ,eiφ) 2
| |
where p C[z,w] is stable, i.e. has no zeros in the closed bidisk D2 =
∈
D D. The characterization is in terms of trigonometric moments of
×
the measure
dθdφ
(2π)2t(θ,φ)
on [0,2π]2, and the necessary and sufficient conditions for the charac-
terization come from studying measures on T2 = (∂D) (∂D) of the
×
form
dz dw
| || |
(2π)2 p(z,w) 2
| |
where p is a polynomial. These are called Bernstein-Szego˝ measures.
In one variable, such measures play a natural role since they can be
used to match a finite sequence of moments of a given positive Borel
measure in an “entropy maximizing” way—see Landau [20] or Simon
[23].
Surprisingly, the development of the above result passes through a
sums ofsquares formula relatedtop which yields afamousinequality of
AndoˆfrommultivariableoperatortheoryandinturnyieldsAgler’sPick
interpolation theorem for bounded analytic functions on the bidisk.
POLYNOMIALS WITH NO ZEROS ON A FACE OF THE BIDISK 3
This connection is described in Cole-Wermer [7] and Knese [16]. The
Hilbert space geometry approach of [16] made it possible to extend the
characterization to the setting where p has no zeros on the open bidisk
D2 in [18].
In another direction, an investigation was begun in [11] of orthogo-
nal polynomials associated with bivariate measures supported on the
bicircle constructed using the lexicographical or reverse lexicographical
ordering and the recurrence formulas associated with these polynomi-
als were developed. As in the one variable case a spectral theory type
result was proved relating the vanishing of certain coefficients in the
recurrence formulas to the existence of a Fej´er-Riesz type factorization
and of a Bernstein-Szego˝ measure with p a stable polynomial. Re-
cently in [8], this viewpoint yielded extensions of the above results to
the problem of characterizing positive bivariate trigonometric polyno-
mials that can be factored as in (1.1) where now p C[z,w] has no
∈
zeros on a closed face of the bidisk. A closed face of the bidisk refers
to either T D or D T. This result is significantly more difficult
× ×
because much of the analyticity of 1/p is lost. However the moments
can still be computed using the one variable residue theorem. Further-
more the factorization is in general not of the Helson-Lowdenslager
type [12] which would give a rational function rather than a polyno-
mial factorization. Special consideration was given when the trigono-
metric polynomial t(θ,φ) = p(z,w) 2 = q(z,w) 2 where p(z,w) = 0
| | | | 6
for z = 1, w 1 whereas q(z,w) = 0 for w = 1, z 1, for
| | | | ≤ 6 | | | | ≤
which a spectral theory result analogous to the characterization of the
Bernstein-Szego˝ measures on the circle was shown to hold.
In this article we refine, extend, and give a more complete picture
of the results in [8]; the case where p has no zeros on a closed face of
the bidisk mentioned above. In particular we emphasize that positive
linear forms on bivariate Laurent polynomials of bounded degree
T
which can be represented as a Bernstein-Szego˝ measure as above with
p(z,w) = 0 for z = 1, w 1 can be characterized in two differ-
6 | | | | ≤
ent ways by using to define an inner product on polynomials: (1)
T
a matrix condition involving certain natural truncated shift operators,
and (2) the existence of a special orthogonal decomposition of spaces of
polynomials we call the split-shift orthogonality condition. The “matrix
condition” is easier to verify (and matches the condition presented in
[8] when we choose an appropriate basis) while the “split-shift orthog-
onality condition” provides more information about the geometry of
the spaces involved as well as the polynomial p. In particular, this lat-
ter condition is key to proving a generalization of the sums of squares
formula alluded to above. A subtle fact is that the spaces involved in
4 J. S.GERONIMO,P. ILIEV,AND G. KNESE
the split-shift condition are in general not unique but are in one-to-
one correspondence with Fej´er-Riesz type factorizations of the positive
trigonometric polynomial t. Furthermore given p we present an explicit
description of these spaces in terms of the decomposition of p(z,0) as
a product of stable and unstable factors. This makes it possible to
characterize a whole stratification of factorizations of t as p 2 where p
| |
has no zeros on T D and p(z,0) has a specified number of zeros in
D. The case where×p(z,0) has no zeros in D recovers the Geronimo-
Woerdeman characterization result, and the case where p(z,0) has all
zeros in D results in a related characterization of when t = p 2 where p
| |
has no zeros in(C D) D. In between these two extremes we can char-
acterize when t =\p 2×where the zero set of p in D C has a specified
number of sheets o|v|er z D sitting in D D (and×a complementary
∈ ×
number of sheets sitting in D (C D)).
× \
We proceed as follows. In Section 2 we introduce the notation used
throughout the paper and state the main theorems. In Section 3 we
derive basic orthogonality relations associated with Bernstein-Szego˝
measures. In Section 4 we show that a Bernstein-Szego˝ measure with
p(z,w) = 0 for z = 1, w 1 implies the split-shift condition using
6 | | | | ≤
a decomposition of p(z,0) into stable and unstable factors. In Sec-
tion 5 we show that the split-shift condition implies the existence of
a Bernstein-Szego˝ measure of the type given above. Next in Section
6 the matrix condition mentioned above is shown to be equivalent to
the split-shift condition. In Section 7 we describe all p that give rise to
the same positive bivariate trigonometric polynomial. In Section 8 we
show how to construct p from the moments associated with the posi-
tive linear form. In Section 9 we apply the previous results to solve the
problem when an extended bivariate autoregressive model has a causal
or acausal solution. Also in this section we give necessary and suffi-
cient conditions in terms of moments when a bivariate Borel measure
supported on the bicircle is a Bernstein-Szego˝ measure with p nonzero
for z = 1, w 1. Finally in Section 10 we adapt ideas from [19]
| | | | ≤
to give a second proof of our generalized sum of squares formula which
should be of independent interest, while we also consider “generalized
distinguished varieties” and apply an argument of adapted from [17] to
obtain a sum of squares formula for polynomials associated with these
varieties. This allows us to obtain a determinantal representation of
the polynomial giving rise to the variety. Distinguished varieties were
introduced in [2] and play an important role in multivariable operator
theory and function theory on the bidisk. Our determinantal represen-
tation generalizes one of the main theorems of [2].
POLYNOMIALS WITH NO ZEROS ON A FACE OF THE BIDISK 5
2. Notation and statement of results
We denote spaces of Laurent polynomials by
= zswt : j s j, k t k ,
j,k
L ∨{ − ≤ ≤ − ≤ ≤ }
where denotes the complex linear span of a set, and we denote spaces
∨
of polynomials by
:= zswt : 0 s j,0 t k ,
j,k
P ∨{ ≤ ≤ ≤ ≤ }
where j,k Z . In some parts of the paper, the spaces and
+ j,k j,k
appear nat∈urally within the context of the Hilbert spacLe L2(T2,Pµ),
where µ is a positive Borel measure on T2, in which case we may use
also j,k = by considering the closed linear spans above.
A linear∞form : C is said to be positive if
n,m
T L →
¯
(f(z,w)f(1/z,1/w)) > 0
T
¯
forevery nonzero f , where f(z,w) = f(z¯,w¯). With we define
n,m
∈ P T
an inner product on the space , via
n,m
P
f,g = (f(z,w)g¯(1/z,1/w)) f,g .
T n,m
h i T ∈ P
Let us write for the finite dimensional Hilbert space ( , , ).
T n,m T
For (k,l) HZ2 where the inner product above is definePd wehd·e·niote
∈ +
the following orthogonal complements:
(2.1) 1 = w ,
Ek,l Pk,l ⊖ Pk,l−1
(2.2) 1 = ,
Fk,l Pk,l ⊖Pk,l−1
(2.3) 2 = z ,
Ek,l Pk,l ⊖ Pk−1,l
(2.4) 2 = .
Fk,l Pk,l ⊖Pk−1,l
←
We will often employ the anti-unitary reflection operator
·
g(z,w) ←g(z,w) := zkwlg¯(1/z,1/w)
7→
which in this case we say is applied at the degree (k,l). This degree
will usually be clear from context or explicitly stated. For example,
applying this operator at degree (k,l) to the spaces 1 and 2 we
Fk,l Fk,l
see that
← ←
1 = 1 and 2 = 2 ,
Ek,l Fk,l Ek,l Fk,l
since the operator is an anti-unitary in .
T
H
Definition 2.1. A positive linear form on satisfies the split-
n,m
T L
shift orthogonality condition if there exist subspaces of polynomials
, such that
1 2 T
K K ⊂ H
(1) 1 =
En−1,m K1 ⊕K2
(2) z and
1 2
K ⊥ K
6 J. S.GERONIMO,P. ILIEV,AND G. KNESE
(3) ,z 1 .
K1 K2 ⊂ En,m
The point of conditions (2) and (3) is that they imply z
1 2
K ⊕ K ⊂
1 . This condition actually characterizes positive linear forms com-
En,m
ing from a Bernstein-Szego˝ measure. What is interesting is that this
condition can also be expressed using a simple matrix condition.
To present the matrix condition let us define three operators
A = P M : 1 w 2
wEn2,m−1 z En−1,m → En,m−1
B = P : w 2 1
En1−1,m Fn,m−1 → En−1,m
T = P M : 1 1
En1−1,m z En−1,m → En−1,m
where M is multiplication by z and P represents orthogonal projec-
z H
tion onto a subspace . Notice that T is just truncation of
T
H ⊂ H
multiplication by z to 1 .
En−1,m
Theorem 2.2. Let be a positive linear form on . The following
n,m
T L
are equivalent.
(1) (Bernstein-Szeg˝o condition) There exists p C[z,w] with no
∈
zeros on T D and degree at most (n,m) such that
×
dz dw
(2.5) (zjwk) = zjwk | || | j n, k m.
T (2π)2 p(z,w) 2 | | ≤ | | ≤
T2
Z | |
(2) (Split-shift condition) satisfies the split-shift orthogonality
T
condition.
(3) (Matrix condition) The invariant subspace of T generated by the
range of B is contained in the kernel of A. More concretely,
(2.6) ATjB = 0 for j = 0,1,...,n 1.
−
ThistheoremisamoregeometricformulationoftheresultsinGeronimo-
Iliev [8]. In particular, the coordinate free formulation of condition (3)
makes it possible to give a straightforward proof of the equivalence of
(2)and(3)inPropositions6.1and6.2—theoriginalproofin[8]involves
some non-trivial linear algebra. Of greater significance, however, is our
emphasis on the split-shift condition and the rather complete knowledge
it provides of the geometry of Bernstein-Szeg˝o measures of the above
type. A version of the split-shift condition was recognized as an im-
portant stepping stone in [8], but at that time it was not clear how to
construct the spaces involved directly with Hilbert space geometry—
this question was explicitly raised as [8, Remark 5.3]. The approach
developed here resolves this.
POLYNOMIALS WITH NO ZEROS ON A FACE OF THE BIDISK 7
Theorem 2.3. Let be a positive linear form on satisfying the
n,m
T L
Bernstein-Szeg˝o condition of Theorem 2.2 with polynomial p(z,w) hav-
ing no zeros on T D and degree at most (n,m). Then
×
= P g(z)a(z) : g C[z],degg < degb
K1 En1−1,m{ ∈ }
= P g(z)b(z) : g C[z],degg < n degb ,
K2 En1−1,m{ ∈ − }
satisfy the split-shift condition. Here p(z,0) = a(z)b(z) where a C[z]
∈
has no zeros in D and b C[z] has all zeros in D.
∈
More explicitly, if we form the span of the following projections of
one variable polynomials
zia(z) P zia(z) for 0 i < degb
− wPn−1,m ≤
and
zib(z) P zib(z) for 0 i < n degb
− wPn−1,m ≤ −
then the resulting subspaces satisfy all the orthogonality conditions in
the split-shift definition.
Why should we emphasize the abstract looking split-shift condition
inthefirst place? Oneanswer tothis isthat the spaces inthe split-shift
condition appear naturally in the following sum of (hermitian) squares
result that ends up being an important by-product of our work.
Theorem 2.4. Suppose p C[z,w] has no zeros on T D and degp =
∈ ×
(n,m). Define p←(z,w) = znwmp(1/z¯,1/w¯). Then, there exist polyno-
mials A ,...,A ,B ,...,B ,C ,...,C C[z,w] such that
1 m 1 n1 1 n2 ∈
p(z,w) 2 p←(z,w) 2
| | −| |
m n1 n2
= (1 w 2) A (z,w) 2 +(1 z 2) B (z,w) 2 C (z,w) 2
j j j
−| | | | −| | | | − | |
!
j=1 j=1 j=1
X X X
where n is the number of zeros of p(z,0) in D and n = n n . The
2 1 2
same result holds if p has no zeros in T D and no factors i−n common
← ×
with p.
The different sums of squares terms can be constructed from impor-
tant subspaces of L2(|dz||dw|) : the A form an orthonormal basis of
|p|2 j
←
2 , the B form an orthonormal basis of (the reflection of ),
En,m−1 j K2 K2
and the C form an orthonormal basis of . (See Theorem 5.5.) This
j 1
K
formula illustrates how natural are the spaces in the split-shift condi-
tion, and it also reproves some important formulas as special cases.
When n = 0, p is stable and we get the Cole-Wermer type of sum
2
of squares formula [7] which can be used to prove Agler’s Pick interpo-
lation theorem on the bidisk; see also [10], [16], [9], [19], [6]. The exact
8 J. S.GERONIMO,P. ILIEV,AND G. KNESE
numbers of squares involved in this case turned out to be important in
recent work on extending Lo¨wner’s theory of matrix monotone func-
tions to two variables in Agler-McCarthy-Young [5]. When m = 0 (i.e.
p does not depend on w) we get a decomposition which readily implies
part of the Schur-Cohn method for counting the roots of a polynomial
inside and outside the unit circle.
The case of p with merely no zeros on T D can be derived from
×
a limiting argument as in [16]. We give a second proof of the sum of
squaresformulausingideasofKummert[19]inSection10.1. Thisproof
should be of independent interest and has the advantage of working di-
rectly for all cases. See Section 10.2 for anapplication of the formula to
provingadeterminantal representation foraclassofcurvesgeneralizing
the distinguished varieties of Agler-McCarthy [2].
Now that we see that the split-shift condition is natural, we get
into a deeper discussion of Theorem 2.2, and its extensions. From the
maximum entropy principle there are at most finitely many p(z,w) for
whichtheBernstein-Szego˝conditionholds. Theorem2.3thereforegives
one particular way to construct , in the definition of split-shift
1 2
K K
and this way is uniquely determined by the choice of p. However, since
a trigonometric polynomial factored as p 2 can potentially be factored
| |
inmorethanonesuch way—roughlyspeaking thesepolynomialscanbe
obtained from one another by permuting the factors in p(z,w) 2 which
| |
depend only on z—each such factorization will yield spaces as in the
split-shift condition via the above theorem. We prove that these are
all the possible split-shift decompositions corresponding to p(z,w) 2.
| |
See Proposition 7.3.
While each choice of p in the factorization of t = p 2 yields a canon-
| |
ically associated pair of spaces , in the split-shift condition, the
1 2
K K
matrix condition naturally gives rise to two canonical choices for such
pairs.
Theorem 2.5. Let be a positive linear form on satisfying the
n,m
T L
matrix condition of Theorem 2.2. Then, ( , ) = ( 1 , )
K1 K2 En−1,m ⊖ B B
satisfies the split-shift condition where
= TjBf : f w 2 ,j = 0,1,... .
B ∨{ ∈ Fn,m−1 }
Similarly, ( , ) = ( , 1 ) satisfies the split-shift condi-
K1 K2 A En−1,m ⊖A
tion where
= (T∗)jA∗f : f w 2 ,j = 0,1,... .
A ∨{ ∈ En,m−1 }
If ( ′, ′) is any other pair satisfying the split-shift condition, then
K1 K2
′ and ′.
A ⊂ K1 B ⊂ K2
POLYNOMIALS WITH NO ZEROS ON A FACE OF THE BIDISK 9
To be clear, T∗ : 1 1 is given by P M and A∗ :
En−1,m → En−1,m En1−1,m 1/z
w 2 1 is given by P M . See Theorems 7.4 and 7.5
En,m−1 → En−1,m En1−1,m 1/z
where we also show how the spaces in Theorem 2.5 relate to those in
Theorem 2.3.
Theorems 2.3 and 2.5 directly show how the Bernstein-Szego˝ con-
dition and the matrix condition yield the split-shift condition. On
the other hand, if the split-shift condition holds, z has co-
1 2
K ⊕ K
dimension one in 1 and we shall show that the Bernstein-Szego˝
En,m
condition holds using any unit norm element p in the one dimensional
space 1 ( z ). A sum of squares result related to Theorem
En,m ⊖ K1 ⊕ K2
2.4 ends up being crucial here. In Section 8, we describe a simple pro-
cedure for constructing p from the moments (zjwk) once we know the
T
split-shift condition holds.
Our emphasis on the split-shift condition permits several interesting
refinements that were not evident before. Notice that if p does not
vanish on T D, then the argument principle shows that the number
of zeros of p×( ,w) in D will be constant as w varies in D. Thus it
·
is possible to prove a “stratified” version of Theorem 2.2, where we
characterize factorizations involving p with no zeros in T D such that
p(z,0) has a specified number of zeros in D. See the end×of Section 7
for the proof of the following corollary.
Corollary 2.6. Let be a positive linear form on and let 0
n,m
T L ≤
d n. The following are equivalent.
≤
(1) (Bernstein-Szeg˝o condition) There exists p C[z,w] with no
∈
zeros on T D, degree at most (n,m), and where p(z,0) has d
zeros in D s×uch that
dz dw
(zjwk) = zjwk | || | j n, k m.
T (2π)2 p(z,w) 2 | | ≤ | | ≤
T2
Z | |
(2) (Split-shift condition) satisfies the split-shift orthogonality
T
condition where has dimension d.
1
K
(3) (Matrix condition) The invariant subspace of T generated by
the range of B is contained in the kernel of A, and
dim d n dim .
A ≤ ≤ − B
Note and are as in Theorem 2.5.
A B
In particular, the case d = 0 yields the Geronimo-Woerdeman result
(aswell asmuchsimpler lookingconditions). Inthiscase, thesplit-shift
condition merely says
(2.7) z 1 1 .
En−1,m ⊂ En,m
10 J. S.GERONIMO,P. ILIEV,AND G. KNESE
Thematrixconditioninthiscaseimplies = 0 which implies A = 0.
A { }
Since the range of A is P z 1 , this means w 2 z 1 ,
wEn2,m−1 En−1,m En,m−1 ⊥ En−1,m
which is equivalent to (2.7) because of the orthogonal decomposition
1 w 2 = z 1 2 .
En,m⊕ En,m−1 En−1,m ⊕En,m
By performing the reflection operation, w 2 z 1 is equiva-
En,m−1 ⊥ En−1,m
lent to 1 2 .
Fn−1,m ⊥ Fn,m−1
Corollary 2.7 (Geronimo-Woerdeman [10]). Let be a positive linear
T
form on . There exists p C[z,w] with no zeros on D2 and degree
n,m
L ∈
at most (n,m) such that
dz dw
(zjwk) = zjwk | || | j n, k m,
T (2π)2 p(z,w) 2 | | ≤ | | ≤
T2
Z | |
if and only if
1 2 .
Fn−1,m ⊥ Fn,m−1
To use the language of [21], the last condition can be neatly phrased
as saying and intersect at right angles.
n−1,m n,m−1
P P
As in [8], Theorem 2.2 allows us to characterize when a positive two
variable trigonometric polynomial can be factored as p(z,w) 2 on T2
| |
where p has no zeros in T D.
×
Theorem2.8. Supposet(z,w) = n m t zjwk > 0 for(z,w)
j=−n k=−m jk ∈
T2. Then, there exists p C[z,w] of degree at most (n,m) with no ze-
∈ P P
ros on T D such that t = p 2 on T2 if and only if the positive linear
× | |
form on
n,m
T L
dz dw
(zjwk) = zjwk | || | j n, k m
T (2π)2t(z,w) | | ≤ | | ≤
T2
Z
satisfies the split-shift condition.
See the end of Section 5 for a proof of this theorem.
We say a finite, positive Borel measure µ on T2 is non-degenerate if
f 2dµ > 0,
| |
T2
Z
for every nonzero polynomial f C[z,w]. We next turnto the problem
of characterizing which such me∈asures µ on T2 are of the form
1
(2.8) dσ
p 2
| |
where p C[z,w] has no zeros in T D and degree at most (n,m); dσ
denotes n∈ormalized Lebesgue measu×re on T2.